Problem: In the diagram, what is the value of $x$? [asy]
size(120);
draw(Circle((0,0),1));
draw((0,0)--(.5,sqrt(3)/2));
draw((0,0)--(sqrt(3)/2,.5));
draw((0,0)--(sqrt(3)/2,-.5));
draw((0,0)--(-1,0));
label("$4x^\circ$",(0,0),NNW); label("$5x^\circ$",(0,0),SSW);
label("$2x^\circ$",(.3,0));label("$x^\circ$",(.3,.3));

[/asy]
The sum of the angles around any point is $360^\circ$. Therefore, $5x^\circ+4x^\circ+x^\circ+2x^\circ=360^\circ$ or $12x = 360$ or $x=\boxed{30}$.